Skepticism about the claims of instrument variable validity/exclusion through a statistical test—the Arellano-Bond Test

I am an applied researcher and occasionally come across papers that have panel data and that use dynamic models with both a fixed-effects term and lagged DV (or multiple autoregressive terms):

$y_{it} = beta_0 + B_1X_{it}+alpha y_{i(t-1)}+delta D_{it} + lambda_i + gamma_t + epsilon_{it}$

where $i$ denotes the panel unit and $t$ denotes the time dimension. The parameter of interest is $delta$ and $D_{it}$ denotes a binary treatment. When the number of time periods is small, such a model cannot be estimated using OLS because of Nickell’s bias.

One approach I have seen people use is to employ higher lags as instruments. The identifying assumption is usually stated as no serial correlation between higher-order error terms.

Is it correct to take this assumption of no serial correlation as the exclusion restriction, i.e., the IV affects the final outcome only through the instrumented variable? If yes, then how does this square with the general point that causality/exclusion cannot generally be established with statistical tests such as the Arellano Bond Test, which statistically tests for the null hypothesis of "no autocorrelation," and proceeds if there is a failure to reject the null for higher orders?

In Mostly Harmless Econometrics (book), Angrist & Pischke write (p. 245):

The problem here is that the differenced residual, $Delta epsilon_{it}$, is necessarily correlated with the lagged dependent variable, $Delta Y_{i(t-1)}$, because both are a function of $epsilon_{i(t-1)}$. Consequently, OLS estimates of (5.3.6) are not consistent for the parameters in (5.3.5), a problem first noted by Nickell (1981). This problem can be solved, though the solution requires strong assumptions. The easiest solution is to use $Y_{i(t-2)}$ as an instrument for $Delta Y_{i(t-1)}$ in (5.3.6).10 But this requires that $Y_{i(t-2)}$ be uncorrelated with the differenced residuals, $Delta epsilon_{it}$. This seems unlikely, since residuals are the part of earnings left over after accounting for covariates. Most people’s earnings are highly correlated from one year to the next, so that past earnings are also likely to be correlated with $Delta epsilon_{it}$. If $epsilon_{it}$ is serially correlated, there may be no consistent estimator for (5.3.6).

Angrist & Pischke make no reference to the Arellano Bond Test to establish the validity/exclusion of the IV. Instead, they make qualitative arguments as I generally see with IV models used for other types of data generation processes.

Does the Arellano Bond (AB) Test really establish exclusion/validity? Or, is it merely a diagnostic that may be used as a secondary argument along with primarily qualitative arguments for exclusion. If the AB test is merely a diagnostic, how should one evaluate research studies that assert identification on the basis of the AB test? (i.e., the AB test fails to reject the null of "no autocorrelation" but qualitatively, one may have reasons to believe that there should be a correlation but the current sample does not show it).

NOTE: Slightly edited version cross-posted on https://stats.stackexchange.com/questions/490747/skepticism-about-the-claims-of-instrument-variable-validity-exclusion-through-a